For optical triangulation systems, the accuracy of the range data
depends on proper interpretation of imaged light reflections. The
most common approach is to reduce the problem to one of finding the
``center'' of a one dimensional pulse, where the ``center'' refers to
the position on the sensor which hopefully maps to the center of the
illuminant. Typically, researchers have opted for a statistic such as
mean, median or peak of the imaged light as representative of the
center. These statistics give the correct answer when the
surface is perfectly planar, but they are generally inaccurate
whenever the surface perturbs the shape of the illuminant.

In Figure 2, we give
examples of how the first three circumstances result in range errors
even for an ideal triangulation system with infinite sensor resolution
and perfect calibration. For purposes of illustration, we omit
the imaging optics of Figure 1 and treat the
sensor as a one dimensional orthographic sensor. Further, we assume
an illuminant of Gaussian cross-section, and we use the mean for
determining the center of an imaged pulse.
Figure 2a shows how a step reflectance
discontinuity results in range points that do not lie on the surface.
Figure 2b and 2c provide two
examples of shape variations resulting in range errors. Note that in
Figure 2c, the center of the illuminant is not
even striking a surface. In this case, a measure of the center of the
pulse results in a range value, when in fact the correct answer is to
return no range value whatever. Finally,
Figure 2d shows the effect of occluding the line
of sight between the illuminated surface and the sensor. This range
error is very similar to the error encountered in
Figure 2c.

The fourth source of range error is laser speckle, which arises when
coherent laser illumination bounces off of a surface that is rough
compared to a wavelength [7]. The
surface roughness introduces random variations in optical path
lengths, causing a random interference pattern throughout space and at
the sensor. The result is an imaged pulse with a noise component
that affects the mean pulse detection, causing range errors even from
a planar target.

To quantify the errors inherent in using mean pulse analysis, we have
computed the errors introduced by reflectance and shape variations for
an ideal triangulation system with a single Gaussian illuminant. We
take the beam width, w, to be the distance between the beam center
and the point of the irradiance profile, a convention common
to the optics literature. We present the range errors in a scale
invariant form by dividing all distances by the beam width.
Figure 3 illustrates the maximum deviation from
planarity introduced by scanning reflectance discontinuities of
varying step magnitudes for varying triangulation angles. As the size
of the step increases, the error increases correspondingly. In
addition, smaller triangulation angles, which are desirable for
reducing the likelihood of missing data due to sensor occlusions,
actually result in larger range errors. This result is not
surprising, as sensor mean positions are converted to depths through a
division by , where is the triangulation angle, so
that errors in mean detection translate to larger range errors for
smaller triangulation angles.

Figure 4 shows the effects of a corner on
range error, where the error is taken to be the shortest distance
between the computed range data and the exact corner point. The
corner is oriented so that the illumination direction bisects the
corner's angle as shown in Figure 2b. As we
might expect, a sharper corner results in greater compression of the
left side of the imaged Gaussian relative to the right side, pushing
the mean further to the right on the sensor and pushing the
triangulated point further behind the corner. In this case, the
triangulation angle has little effect as the division by
is offset almost exactly by the smaller observed left/right pulse
compression imbalance.

One possible strategy for reducing these errors would be to decrease
the width of the beam and increase the resolution of the sensor.
However, diffraction limits prevent us from focusing the beam to an
arbitrary width. The limits on focusing a Gaussian beam with
spherical lenses are well known [15]. In
recent years, Bickel, et al, [3]
have explored the use of axicons (e.g., glass cones and other surfaces
of revolution) to attain tighter focus of a Gaussian beam. The
refracted beam, however, has a zeroth order Bessel function
cross-section; i.e., it has numerous side-lobes of non-negligible
irradiance. The influence of these side-lobes is not well-documented
and would seem to complicate triangulation.